Full Answer Section

 

 

 

 

• Q3 (Third Quartile): This is the median of the upper half of the data. The upper half contains the last 22 values. The median of these 22 values is the average of the (23 + 11)th and (23 + 12)th values, which are the 34th and 35th values: (139 + 140) / 2 = 139.5.
• IQR (Interquartile Range): This is the difference between the third quartile (Q3) and the first quartile (Q1): IQR = Q3 – Q1 = 139.5 – 102 = 37.5.

Box-plot Construction (Conceptual):

1. Draw a number line that includes the range of the data (from 30 to 206).
2. Draw a box extending from Q1 (102) to Q3 (139.5).
3. Draw a vertical line inside the box at the median (114).
4. Calculate the upper and lower fences to identify potential outliers:
   • Lower Fence = Q1 – 1.5 * IQR = 102 – 1.5 * 37.5 = 102 – 56.25 = 45.75
   • Upper Fence = Q3 + 1.5 * IQR = 139.5 + 1.5 * 37.5 = 139.5 + 56.25 = 195.75
5. Draw whiskers extending from the box to the smallest data point within the lower fence (87) and the largest data point within the upper fence (180).
6. Any data points outside the fences (30 and 206) would be plotted as individual points, indicating potential outliers.

Question 2: Sports Car Weights

a. Find the mean of the sample weights. Round to two decimal places.

To find the mean, we sum all the weights and divide by the number of cars (30).

Sum of weights = 3959 + 2868 + 3577 + 2606 + 3956 + 3384 + 3393 + 3679 + 3867 + 2578 + 2523 + 3834 + 2578 + 3326 + 3504 + 2841 + 2713 + 3457 + 3059 + 2959 + 3714 + 2554 + 3095 + 3771 + 2542 + 3425 + 3942 + 3897 + 3474 + 3594 = 101678

Mean (x̄) = Sum of weights / Number of cars = 101678 / 30 = 3389.27 (rounded to two decimal places).

b. Find the standard deviation of the sample weights. Round to two decimal places.

To find the sample standard deviation (s), we use the formula:

s=n−1∑(xi​−xˉ)2​<img class=”katex-svg” src=”http://superioressaywriters.com/data:;base64,​

Where:

• xi​ is each individual weight
• xˉ is the mean (3389.27)
• $n$ is the number of cars (30)

We need to calculate the squared difference between each weight and the mean, sum these squared differences, divide by (n-1), and then take the square root.

This is a more involved calculation. Using a calculator or statistical software, the sample standard deviation is approximately 480.49 (rounded to two decimal places).

c. Construct one standard deviation of the mean.

One standard deviation of the mean refers to the interval within one standard deviation above and below the mean.

Lower bound = Mean – Standard Deviation = 3389.27 – 480.49 = 2908.78 Upper bound = Mean + Standard Deviation = 3389.27 + 480.49 = 3869.76

The interval within one standard deviation of the mean is (2908.78, 3869.76).

d. What percent of the truck weights were within one standard deviation of the mean?

Now we count how many of the 30 weights fall within the interval (2908.78, 3869.76):

3959 (No) 2868 (No) 3577 (Yes) 2606 (No) 3956 (No) 3384 (Yes) 3393 (Yes) 3679 (Yes) 3867 (Yes) 2578 (No) 2523 (No) 3834 (Yes) 2578 (No) 3326 (Yes) 3504 (Yes) 2841 (No) 2713 (No) 3457 (Yes) 3059 (Yes) 2959 (Yes) 3714 (Yes) 2554 (No) 3095 (Yes) 3771 (Yes) 2542 (No) 3425 (Yes) 3942 (No) 3897 (No) 3474 (Yes) 3594 (Yes)

Number of weights within one standard deviation = 18

Percentage = (Number of weights within one standard deviation / Total number of weights) * 100 Percentage = (18 / 30) * 100 = 60%

Question 3: House of Representatives Men

a. Produce a scatterplot of the data. Include the line of best fit on the graph.

To do this accurately, you would need to use a graphing tool or statistical software. The x-axis would represent the “Year Position Began” and the y-axis would represent the “Number of Men”. The line of best fit (also called the trend line or regression line) would be a straight line that best represents the general trend of the data points.

(Conceptual Description of the Scatterplot):

The scatterplot would show points generally trending downwards from the top left to the bottom right, indicating a negative relationship between the year and the number of men. The line of best fit would be a straight line drawn through the points, attempting to minimize the distance between the line and each point.

b. Give the line of best fit for the data and the correlation coefficient rounded to 4 decimal places. Is the correlation positive, negative, weak, strong?

Using statistical software to perform a linear regression on the data, we would obtain the equation of the line of best fit (in the form y = mx + c, where y is the number of men and x is the year) and the correlation coefficient (r).

(Results from Regression Analysis – Approximate):

• Line of Best Fit (Approximate): y = -4.85x + 9908.79
• Correlation Coefficient (r) (Approximate): -0.9887

Interpretation of the Correlation Coefficient:

• The correlation coefficient is negative because the number of men tends to decrease as the year increases.
• The absolute value of the correlation coefficient (|-0.9887| = 0.9887) is very close to 1. This indicates a strong negative linear correlation between the year and the number of men in the House of Representatives.

c. Is the number of men increasing or decreasing? By how is the number of men changing per year?

• The number of men is decreasing over time, as indicated by the negative correlation and the downward trend in the scatterplot.
• The slope of the line of best fit (m) represents the average change in the number of men per year. In our approximate line of best fit (y = -4.85x + 9908.79), the slope is -4.85. This suggests that, on average, the number of men in the House of Representatives is decreasing by approximately 4.85 men per year.

d. Based on the line of best fit, predict the number of men that will be in the House of Representatives in 2027.

Using the approximate line of best fit equation (y = -4.85x + 9908.79), we can predict the number of men in 2027 by substituting x = 2027:

y = -4.85 * 2027 + 9908.79 y = -9830.45 + 9908.79 y = 78.34

Based on this linear model, we would predict approximately 78 men in the House of Representatives in 2027.

Important Note: This prediction is based on a linear trend and may not perfectly reflect future realities. Political and social factors can influence the composition of the House.